Question

Prove that the function is continuous at

Answer 1

The given function is,

f( x )=5x−3

At x=0, the function becomes,

f( 0 )=0−3 =−3

The left hand limit of the function at x=0 is,

LHL= lim x→ 0 − f( x ) = lim x→ 0 − ( 5x−3 ) =−3

The right hand limit of the function at x=0 is,

RHL= lim x→ 0 + f( x ) = lim x→ 0 + ( 5x−3 ) =−3

It can be observed that, at x=0, LHL=RHL=f( 0 )=−3.

Therefore, the function is continuous at x=0.

At x=−3, the given function becomes,

f( −3 )=5( −3 )−3 =−18

The left hand limit of the function at x=−3 is,

LHL= lim x→ 3 − f( x ) = lim x→ 3 − ( 5x−3 ) =−18

The right hand limit of the function at x=−3 is,

RHL= lim x→ 3 + f( x ) = lim x→ 3 + ( 5x−3 ) =−18

It can be observed that, at x=−3, LHL=RHL=f( −3 )=−18

Therefore, the function is continuous at x=−3.

At x=5, the given function becomes,

f( 5 )=5( 5 )−3 =22

The left hand limit of the function at x=5 is,

LHL= lim x→ 5 − f( x ) = lim x→ 5 − ( 5x−3 ) =22

The right hand limit of the function at x=5 is,

RHL= lim x→ 5 + f( x ) = lim x→ 5 + f( 5x−3 ) =22

It can be observed that, at x=5, LHL=RHL=f( 5 )=22.

Therefore, the function is continuous at x=5.